Solve \int_1^e ((lnx)^(5))/(x)dx

Evaluate

\int_{1}^{e} \frac{( ln ( x ) ) ^{5}}{x} d x

Evaluate the integral

\int \frac{( ln ( x ) ) ^{5}}{x} d x

Use the substitution d x = x d t to transform the integral

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\int \frac{( ln ( x ) ) ^{5}}{x} \times x d t

Simplify

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\int (ln (x))^{5} d t

Use the substitution t = ln (x) to transform the integral

\int t^{5} d t

Use the property of integral \int x^{n} d x = \frac{x ^{n + 1}}{n + 1}

\frac{t ^{5 + 1}}{5 + 1}

Add the numbers

\frac{t ^{6}}{5 + 1}

Add the numbers

\frac{t ^{6}}{6}

Substitute back

\frac{( ln ( x ) ) ^{6}}{6}

Simplify the expression

\frac{1}{6} (ln (x))^{6}

Return the limits

(\frac{1}{6} (ln (x))^{6})_{1}^{e}

Solution

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\frac{1}{6}

Alternative Form

0.1 \dot{6}